Arrested Bubble Rise in a Narrow Tube

If a long air bubble is placed inside a vertical tube closed at the top it can rise by displacing the fluid above it. However, Bretherton found that if the tube radius, R, is smaller than a critical value Rc=0.918ℓc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{c}=0.918 \; \ell _c$$\end{document}, where ℓc=γ/ρg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _c=\sqrt{\gamma /\rho g}$$\end{document} is the capillary length, there is no solution corresponding to steady rise. Experimentally, the bubble rise appears to have stopped altogether. Here we explain this observation by studying the unsteady bubble motion for R<Rc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R<R_{c}$$\end{document}. We find that the minimum spacing between the bubble and the tube goes to zero in limit of large t like t-4/5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^{-4/5}$$\end{document}, leading to a rapid slow-down of the bubble’s mean speed U∝t-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U \propto t^{-2}$$\end{document}. As a result, the total bubble rise in infinite time remains very small, giving the appearance of arrested motion.